A note on critical problems involving the $p$-Grushin Operator: existence of infinitely many solutions

TL;DR

The paper addresses a critical boundary-value problem for the $p$-Grushin operator in a bounded domain, featuring a Sobolev-critical nonlinearity. It combines a truncation technique with Krasnoselskii's genus to establish the existence of infinitely many weak solutions for sufficiently small $\lambda>0$. This work extends classical multiplicity results from the standard $p$-Laplacian to the degenerate Grushin framework and hinges on a sharp Palais–Smale condition below an explicit energy threshold, supported by concentration-compactness and genus-based minimax methods. The results advance multiplicity theory for critical PDEs in Grushin-type spaces and provide a blueprint for handling similar degenerate operators via a truncated variational approach.

Abstract

We consider a critical problem in a bounded domain involving the $p$-Grushin operator $Δ_α^p$. After a truncation argument, we obtain infinitely many solutions to our problem via Krasnoselskii's genus, extending a previous result of García Azorero and Peral Alonso to the $p$-Grushin operator. A central part of our analysis is the verification of the Palais-Smale condition of the associated functional under a certain level.

Figures (1)

• Figure 1:

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Theorem 2.1: MMBS25
• Proposition 2.2
• Lemma 2.3
• Remark 2.4
• Lemma 3.1
• proof
• Lemma 4.1